# Dr. Ali Uncu

### Research Area

Integer Partitions, q-Series, Combinatorics### My personal website: www.akuncu.com.

## Ongoing Projects

### Computer Algebra for Nested Sums and Products [FWF SFB F050-09]

## Software

### qFunctions

#### The qFunctions package is a Mathematica package for q-series and partition theory applications.

The qFunctions package by Jakob Ablinger and Ali K. Uncu is a Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-shift equations and recurrences ...

## Publications

### 2021

### qFunctions - A Mathematica package for q-series and partition theory applications

#### J. Ablinger, A. Uncu

Journal of Symbolic Computation 107, pp. 145-166. 2021. ISSN 0747-7171. arXiv:1910.12410. [doi]**article**{RISC6299,

author = {J. Ablinger and A. Uncu},

title = {{qFunctions -- A Mathematica package for q-series and partition theory applications}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {107},

pages = {145--166},

isbn_issn = {ISSN 0747-7171},

year = {2021},

note = {arXiv:1910.12410},

refereed = {yes},

length = {22},

url = {https://doi.org/10.1016/j.jsc.2021.02.003}

}

### 2019

### Polynomial Identities Implying Capparelli's Partition Theorems

#### Ali Kemal Uncu, Alexander Berkovich

Accepted - Journal of Number Theory, pp. -. 2019. N/A. [url]**article**{RISC5790,

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Polynomial Identities Implying Capparelli's Partition Theorems }},

language = {english},

journal = {Accepted - Journal of Number Theory},

pages = {--},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {21},

url = {https://arxiv.org/pdf/1807.10974.pdf}

}

### Refined q-Trinomial Coefficients and Two Infinite Hierarchies of q-Series Identities

#### Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (accepted), pp. 1-10. 2019. N/A. [url]**article**{RISC5801,

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarchies of q-Series Identities }},

language = {english},

abstract = {We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

journal = {ArXiv e-prints (accepted)},

pages = {1--10},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {10},

url = {https://arxiv.org/abs/1810.12048}

}

### A Polynomial Identity Implying Schur's Partition Theorem

#### Ali Kemal Uncu

ArXiv e-prints (submitted), pp. 1-11. 2019. N/A. [url]**article**{RISC5898,

author = {Ali Kemal Uncu},

title = {{A Polynomial Identity Implying Schur's Partition Theorem }},

language = {english},

abstract = {We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and q-series identities. },

journal = {ArXiv e-prints (submitted)},

pages = {1--11},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {11},

url = {https://arxiv.org/abs/1903.01157}

}

### 2018

### On some polynomials and series of Bloch-Polya Type

#### Berkovich A., Uncu A. K.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 146(7), pp. 2827-2838. July 2018. 1088-6826. [url]**article**{RISC5557,

author = {Berkovich A. and Uncu A.~K.},

title = {{On some polynomials and series of Bloch-Polya Type}},

language = {english},

journal = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY},

volume = {146},

number = {7},

pages = {2827--2838},

isbn_issn = {1088-6826},

year = {2018},

month = {July},

refereed = {yes},

keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A17, 05A19, 11B65, 11P81},

length = {12},

url = {http://www.ams.org/journals/proc/2018-146-07/S0002-9939-2018-13982-9/}

}

### Some Elementary Partition Inequalities and Their Implications

#### Berkovich A., Uncu A. K.

ArXiv e-prints (to appear in Annals of Cobinatorics), pp. -. 2018. Preprint. [url]**article**{RISC5558,

author = {Berkovich A. and Uncu A.~K.},

title = {{Some Elementary Partition Inequalities and Their Implications}},

language = {english},

journal = {ArXiv e-prints (to appear in Annals of Cobinatorics)},

pages = {--},

isbn_issn = {Preprint},

year = {2018},

refereed = {yes},

keywords = {Mathematics - Combinatorics, Mathematics - Number Theory, 05A15, 05A17, 05A19, 05A20, 11B65, 11P81, 11P84, 33D15},

length = {12},

url = {https://arxiv.org/abs/1708.01957}

}

### Elementary Polynomial Identities Involving q-Trinomial Coefficients

#### Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (submitted), pp. -. 2018. N/A. [url]**article**{RISC5791,

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Elementary Polynomial Identities Involving q-Trinomial Coefficients }},

language = {english},

journal = {ArXiv e-prints (submitted)},

pages = {--},

isbn_issn = {N/A},

year = {2018},

refereed = {yes},

length = {0},

url = {https://arxiv.org/abs/1810.06497}

}

### On double sum generating functions in connection with some classical partition theorems

#### Ali Kemal Uncu

ArXiv e-prints, pp. 1-20. 2018. N/A.**article**{RISC5800,

author = {Ali Kemal Uncu},

title = {{On double sum generating functions in connection with some classical partition theorems }},

language = {english},

abstract = { We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed here are the set of Rogers--Ramanujan, Göllnitz--Gordon, and little Göllnitz partitions. This work also includes finding the finite analogs of the related generating functions and the discussion of some related series and polynomial identities. Additionally, we present a different construction and a double sum representation for the products similar to the ones that appear in the Rogers--Ramanujan identities. },

journal = {ArXiv e-prints},

pages = {1--20},

isbn_issn = {N/A},

year = {2018},

refereed = {yes},

length = {20}

}

### 2017

### Variation on a theme of Nathan Fine. New weighted partition identities

#### Berkovich Alexander, Uncu Ali K.

J. Number Theory 176, pp. 226-248. 2017. ISSN 0022-314X. [doi]**article**{RISC5552,

author = {Berkovich Alexander and Uncu Ali K.},

title = {{Variation on a theme of Nathan Fine. New weighted partition identities}},

language = {english},

journal = {J. Number Theory},

volume = {176},

pages = {226--248},

isbn_issn = { ISSN 0022-314X},

year = {2017},

refereed = {yes},

length = {23},

url = {https://doi.org/10.1016/j.jnt.2016.12.011}

}

### Weighted Rogers-Ramanujan partitions and Dyson crank

#### Uncu Ali Kemal

The Ramanujan Journal, pp. -. May 2017. ISSN 1572-9303. [doi]**article**{RISC5555,

author = {Uncu Ali Kemal},

title = {{Weighted Rogers--Ramanujan partitions and Dyson crank}},

language = {english},

abstract = {In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts' sum is n is equal to the number of partitions of n with non-negative crank.},

journal = {The Ramanujan Journal},

pages = {--},

isbn_issn = { ISSN 1572-9303},

year = {2017},

month = {May},

refereed = {yes},

length = {0},

url = {https://doi.org/10.1007/s11139-017-9903-8}

}

### 2016

### On partitions with fixed number of even-indexed and odd-indexed odd parts

#### Berkovich Alexander, Uncu Ali Kemal

J. Number Theory 167, pp. 7-30. 2016. ISSN 0022-314X. [doi]**article**{RISC5553,

author = {Berkovich Alexander and Uncu Ali Kemal},

title = {{On partitions with fixed number of even-indexed and odd-indexed odd parts}},

language = {english},

journal = {J. Number Theory},

volume = {167},

pages = {7--30},

isbn_issn = { ISSN 0022-314X},

year = {2016},

refereed = {yes},

length = {24},

url = {https://doi.org/10.1016/j.jnt.2016.02.031}

}

### New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions

#### Berkovich A., Kemal Uncu A.

ALLADI60 2016: Analytic Number Theory, Modular Forms and q-Hypergeometric Series , pp. -. 2016. 978-3-319-68375-1. [url]**article**{RISC5556,

author = {Berkovich A. and Kemal Uncu A.},

title = {{New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions}},

language = {english},

journal = {ALLADI60 2016: Analytic Number Theory, Modular Forms and q-Hypergeometric Series },

pages = {--},

isbn_issn = {978-3-319-68375-1},

year = {2016},

refereed = {yes},

keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A15, 05A17, 05A19, 11B34, 11B75, 11P81, 11P84, 33D15},

length = {0},

url = {https://link.springer.com/book/10.1007/978-3-319-68376-8}

}

### 2015

### A new companion to Capparelli's identities

#### Berkovich Alexander, Uncu Ali Kemal

Adv. in Appl. Math. 71, pp. 125-137. 2015. ISSN 0196-8858. [doi]**article**{RISC5554,

author = {Berkovich Alexander and Uncu Ali Kemal},

title = {{A new companion to Capparelli's identities}},

language = {english},

journal = {Adv. in Appl. Math.},

volume = {71},

pages = {125--137},

isbn_issn = { ISSN 0196-8858},

year = {2015},

refereed = {yes},

length = {13},

url = {https://doi.org/10.1016/j.aam.2015.09.012}

}